Polite Requests

Asking people to do things:

Can you wait a moment, please?
Could you tell me how to get to the library?
Do you think you could post my letter?
I wonder if you could sign the paper?
Do you mind helping me with my project?

Asking for and giving permission:

May I come in? "yes, please do."
Could I use your calculator? "yes. sure'
Can I speak to Mr. Drill, please?

For offering an invitation:

Would you like a cup of tea?
Would you like to play chess with me?

Informally to a friend

Do you think you could turn the volume of the TV down?
Do you think you could give me a lift to the station?
Do you think I could borrow your car?

Formally to a Stranger

Could you tell me where the nearest ATM is, please?
Would you mind if I sit here?
Would you mind lending me your book, please?
Would you like to sit here?
Would you mind giving me a hand with my suitcase?

Note:

We can add the phrase by any chance at the end of our request to make it more polite. (More for information requests)